Question Number 199001 by mnjuly1970 last updated on 26/Oct/23
$$ \\ $$$$\:\mathrm{I}{f}\:,\mathrm{A}\:\in\:\mathrm{M}_{{n}×{n}} \:\:\:,\:\:\mathrm{A}^{\:\mathrm{2}} \:=\:\mathrm{A}\:,\mathrm{1}\neq\:{k}\:\in\mathbb{R}. \\ $$$$\:\:\:\mathrm{F}{ind}\:\:\:\:\left(\:\:\:\mathrm{I}\:−\:{k}\mathrm{A}\:\right)^{\:−\mathrm{1}} \:=\:?\: \\ $$
Answered by MM42 last updated on 26/Oct/23
$${if}\:\:\mid{A}\mid\neq\mathrm{0} \\ $$$$\Rightarrow{A}^{\mathrm{2}} ={A}\:\overset{×{A}^{−\mathrm{1}} } {\Rightarrow}\:{A}={I}\: \\ $$$$\Rightarrow\left({I}−{kA}\right)^{−\mathrm{1}} =\left({I}−{kI}\right)^{−\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{1}−{k}}\:{I}\:\checkmark\:\: \\ $$$$ \\ $$